You can stay near the airport if you want, but that won't be very fun. It may save some money, but there are better options. And regardless of whether or not you stay out there, you should spring for the cab ride. There are some shuttle services to different landmarks in the city, like Grand Central Station, Penn Station, Port Authority, etc. that run about $15 per person, or so, and a cab ride will be maybe $40/50 and will make you feel more comfortable, and provide the direct services the shuttles do not.
If you want a hotel closer to the ship, I know there's a place called the Skyline Hotel that's on the west side and as of now it looks like they still have rooms. (skylinehotelny dot com) It's not the most glamorous, but it seems cheaper than the places closer to Times Square, and depending on your luggage, you could walk to the ship. Or it would just be a tiny cab fare because it's so close. Either way, stick to cabs and you'll be happy you did.
Other options for hotels, I'd say, is anything on the Upper West Side. I just did a quick search and Expedia has some really good rates. Do a Google search for "Hotels UWS NYC."
Don't be afraid of the Upper West Side. Or even a little higher up. The area is generally great (even high up) and you'll be also a quick cab ride to the ship and save on lodging.
It seems like most hotels closer to Times Square are booked at that time anyway since it is Labor Day weekend. Book soon!
I think the Skyline Hotel, or anything Upper West Side might be ideal for what you're looking for, though.
. Hoping to feel better soon, and for summer to finally arrive. So, back off to bed for me ... 
